Effect of Higher-Order Nonlinearities on Amplification …epiq.physique.usherbrooke.ca/data/files/publications/JPA...Effect of Higher-Order Nonlinearities on Amplification and Squeezing

Effect of Higher-Order Nonlinearities on Amplification and Squeezingin Josephson Parametric Amplifiers

Samuel Boutin,1,* David M. Toyli,2,3 Aditya V. Venkatramani,2,3,† Andrew W. Eddins,2,3

Irfan Siddiqi,2,3 and Alexandre Blais1,41Institut quantique et Departement de Physique, Universite de Sherbrooke,

Sherbrooke, Quebec J1K 2R1, Canada2Quantum Nanoelectronics Laboratory, Department of Physics, University of California,

Berkeley, California 94720, USA3Center for Quantum Coherent Science, University of California, Berkeley, California 94720, USA

4Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada(Received 7 August 2017; published 15 November 2017)

Single-mode Josephson-junction-based parametric amplifiers are often modeled as perfect amplifiersand squeezers. We show that in practice, the gain, quantum efficiency, and output field squeezing of thesedevices are limited by usually neglected higher-order corrections to the idealized model. To arrive at thisresult, we derive the leading corrections to the lumped-element Josephson parametric amplifier of threecommon pumping schemes: monochromatic current pump, bichromatic current pump, and monochromaticflux pump. We show that the leading correction for the last two schemes is a single Kerr-type quartic term,while the first scheme contains additional cubic terms. In all cases, we find that the corrections aredetrimental to squeezing. In addition, we show that the Kerr correction leads to a strongly phase-dependentreduction of the quantum efficiency of a phase-sensitive measurement. Finally, we quantify the departurefrom the ideal Gaussian character of the filtered output field from numerical calculation of third- andfourth-order cumulants. Our results show that while a Gaussian output field is expected for an idealJosephson parametric amplifier, higher-order corrections lead to non-Gaussian effects which increase withboth gain and nonlinearity strength. This theoretical study is complemented by experimental characteri-zation of the output field of a flux-driven Josephson parametric amplifier. In addition to a measurement ofthe squeezing level of the filtered output field, the Husimi Q function of the output field is imaged by theuse of a deconvolution technique and compared to numerical results. This work establishes nonlinearcorrections to the standard degenerate parametric amplifier model as an important contribution to theJosephson parametric amplifier’s squeezing and noise performance.

DOI: 10.1103/PhysRevApplied.8.054030

I. INTRODUCTION

Driven by the need for fast, high-fidelity single-shotreadout of superconducting qubits, superconducting low-noise microwave amplifiers are the subject of intenseresearch. Following the path of the work of Yurke et al.in the late 1980s [1–3], several designs of Josephson-junction-based parametric amplifiers (JPAs) have beenintroduced [4–11]. In addition to high-fidelity supercon-ducting qubit readout leading to the observation of quan-tum jumps [12,13], this generation of near-quantum-limitedamplifiers have opened up experimental possibilitiessuch as the creation and tomography of squeezed micro-wave light [14–16] and detailed weak measurement experi-ments [17–19]. JPAs are now ubiquitous in current

superconducting circuit experiments, and applications inother research communities are growing [20–23].Depending on their design and operating mode, JPAs

can fall into either of two broad categories of linearamplifiers: phase preserving and phase sensitive [24,25].JPAs in the former category amplify both quadratures ofthe signal, and quantum mechanics put a strict lowerbound on the noise added by this process. On the contrary,JPAs in the latter category can amplify the signal of asingle quadrature without any added noise by proportion-ally attenuating the conjugate quadrature. In other words,a phase-sensitive amplification is a source of squeezedradiation [26,27]. The properties of JPAs as a source ofsqueezed light are, therefore, intimately related to theirnoise properties as a phase-sensitive amplifier. WhileJPAs are usually modeled as quantum-limited amplifiersand, thus, perfect squeezers, experimental results indicatethat nonidealities limit both the achievable level ofsqueezing [28–30] and the measurement quantum effi-ciency [17,29,31,32].

*[email protected]†Present address: Department of Physics, Harvard University,

Cambridge, Massachusetts 02138, USA.

PHYSICAL REVIEW APPLIED 8, 054030 (2017)

2331-7019=17=8(5)=054030(22) 054030-1 © 2017 American Physical Society

https://doi.org/10.1103/PhysRevApplied.8.054030




We show that these nonidealities are linked to higher-order corrections to the JPA Hamiltonian due to theJosephson cosine potential. We go beyond the standardanalytical results by considering numerical solutions tothe quantum master equation, including these usuallyneglected higher-order corrections. We derive the correc-tions to the JPA Hamiltonian for the single-mode andsingle-port lumped-element JPA [7,25]. Using the forma-lism of quantum optics, we compare three frequently usedpumping schemes of the JPA: monochromatic currentpump [7,33], bichromatic current pump [34], and mono-chromatic flux pump [5,35,36]. We derive the leadinghigher-order corrections for each and study numericallytheir effect on gain, quantum efficiency, squeezing level,and Gaussianity of the output field. In addition, bycomparing numerical results to an experimental characteri-zation of the JPA output field, we show that the squeezinglevel saturation previously reported in the literature [28,29]can be explained by including leading nonquadratic cor-rections in the Hamiltonian. The focus of this work onhigher-order corrections complements recent theoreticalinvestigations of various JPA designs [25,33–35,37,38].The paper is organized as follows. In Sec. II, we set the

notations and recall useful results for the standard quantum-optics model of the degenerate parametric amplifier (DPA).In Sec. III, we introduce the lumped-element JPA and thethree different pumping schemes investigated in this work.For each scheme, we derive the higher-order correctionsand show how, in the low-gain and low nonlinearity regime,the system can be mapped back to the DPA. The respectiveadvantages of the three pumping schemes are compared. InSec. IV, we compare the intracavity field properties of thedifferent schemes including deviations from the resultsof the DPA. We also show that higher-order correctionscan lead to non-Gaussian intracavity fields. In Sec. V, wecharacterize the JPA as an amplifier by calculating the gainand the quantum efficiency for both the phase-sensitiveand phase-preserving modes of operation. In order to relatethese results to a series of experiments [28–30], in Sec. VI,we focus on the phase-sensitive amplification of vacuum,i.e., squeezed vacuum. We characterize the JPA as a sourceof squeezed light by calculating moments and cumulantsof the filtered output field. This allows us to obtain thesqueezing level of the light generated, as well as estimate itsnon-Gaussian character. In this section, numerical resultsare discussed together with an experimental characteriza-tion of the output field, including a direct imaging of thenon-Gaussian distortions of the field due to nonidealities.Finally, Sec. VII summarizes our work.

II. DEGENERATE PARAMETRIC AMPLIFIERIN A NUTSHELL

To set the notation, we begin by presenting the DPAmodel and its solution [27,33,39]. In the next sections, weshow how the JPA can be mapped to the DPA and study

deviations from this simple model due to higher-ordercorrections.The DPA is a standard model of quantum optics

exhibiting parametric amplification and squeezing. In thismodel, as illustrated in Fig. 1(a), a nonlinear mediuminserted in a single-mode cavity is pumped in order tomodulate its refractive index at twice the cavity frequency[26]. This modulates the effective length of the cavityand, as a consequence, its frequency. This modulation actsas an external source of energy leading to parametricamplification [40,41].Introducing the annihilation (creation) operator að†Þ for

excitations in the cavity, the DPA Hamiltonian is (settingℏ ¼ 1 for the remainder of the paper)

H ¼ ωca†aþ χða†2ap þ a2a†pÞ; ð1Þ

withωc the cavity frequency, χ the nonlinearity, and að†Þp the

annihilation (creation) operator of an excitation in theexternal pump mode. In the strong classical pump regime,where ap ≈ αpe−iωpt and in a frame rotating at half theparametric pump frequency ωp ∼ 2ωc, the Hamiltonian is[26,42]

HDPA ¼ Δa†aþ λ

2a†2 þ λ�

2a2; ð2Þ

with the detuning Δ ¼ ωc − ωp=2 and λ ¼ 2χαp theamplitude of the parametric pump.Using input-output theory [26,27], the equation of

motion for the intracavity field is

_a ¼ i½HDPA; a� −κ

2aþ ffiffiffi

κp

ain þffiffiffiγ

pbin; ð3Þ

where the input mode ain (coupled to the cavity with rate κ)carries the signal to be amplified, while the input mode bin(coupled to the cavity with rate γ) mixes additional vacuum

(a)

(b)

FIG. 1. (a) Schematic of a DPA as a two-sided cavity in theoptical domain. (b) Circuit of a lossless lumped-element JPA, areflection amplifier in the microwave domain.

SAMUEL BOUTIN et al. PHYS. REV. APPLIED 8, 054030 (2017)

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noise to the signal due to undesired losses. The totaldamping rate of the cavity is given by κ ¼ κ þ γ.As shown in Appendix A, the solution to Eq. (3) can be

obtained in Fourier space. Using the boundary condition[26,27]

aout ¼ffiffiffiκ

pa − ain; ð4Þ

where aout is the output field carrying the amplified signal,one obtains the solution

aout½ω� ¼ gS;ωain½ω� þ gI;ωain†½−ω�

þffiffiffiγ

κ

r½ðgS;ω þ 1Þbin½ω� þ gI;ωbin

†½−ω��; ð5Þ

where the signal and idler amplitude gains are defined as

gS;ω ¼�κκ=2 − iκðΔþ ωÞ

D½ω� − 1

�; ð6Þ

gI;ω ¼ −iκλD½ω� ; ð7Þ

with D½ω� ¼ Δ2 þ ðκ=2 − iωÞ2 − jλj2, and the frequency ωdefined in the rotating frame such that a signal at ω ¼ 0 isin resonance with the rotating-frame frequency ωp=2.The output signal at frequency ω mixes and amplifiesthe input signal and idler modes at frequencies �ω. In thelossless case (γ ¼ 0), the signal and idler gains obeyjgS;ωj2 ¼ jgI;ωj2 þ 1, and the input-output relation is aunitary squeezing transformation [24]. On the contrary,in the presence of losses (γ ≠ 0), additional noise is mixedwith the input signal, and the DPA is not a quantum-limitedamplifier.If the measurement bandwidth includes both the signal

and idler modes, these two modes act effectively as a singlemode, and the DPA is a phase-sensitive amplifier. Onthe contrary, if the idler frequency (−ω) falls outside themeasured frequency band, the idler mode acts as a noisemode, and the DPA is a phase-preserving amplifier withphase-preserving photon-number gain [24,37]

Gω ¼ jgS;ωj2: ð8Þ

Hence, depending on the experimental details, the samesystem can act either as a phase-sensitive or phase-preserving amplifier. The same holds true for the JPA,and we, thus, consider both cases when characterizing theJPA properties as an amplifier in Sec. V.Finally, in both operating regimes, the gain diverges

(D½ω� ¼ 0) at the parametric threshold [35,39]

λcrit ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔ2 þ κ2=4;

qð9Þ

and large gain is obtained near but below this value. Indeed,above the threshold, spontaneous parametric oscillationeffects will occur, leading to the generation of photonsactivated by vacuum and thermal fluctuations [35,43]. Inthiswork,we focus on parameter regimeswhere theDPA actsas an amplifier, and, thus, λ < λcrit is always considered.

III. HIGHER-ORDER CORRECTIONS TO THEJPA: COMPARISON OF PUMPING SCHEMES

In this section, we introduce the standard lumped-element JPA circuit and consider three pumping schemes:the monochromatic current pump, the bichromatic currentpump, and the monochromatic flux pump. We show howthese amplifiers can be mapped back to the DPA model andcompare their respective advantages. Importantly, for eachpumping scheme, we derive the leading nonidealities whichcause deviations from the DPA model. The study of thesedeviations in the following sections constitutes the core ofour results.

A. JPA circuit

As we illustrate in Fig. 1(b), the lumped-element JPA issimply a capacitively shunted Josephson junction coupledto a transmission line [1,7]. The Hamiltonian of thisstandard circuit is

HJPA ¼ Q2

2C− EJ cos

�ϕ

φ0

�; ð10Þ

with C the capacitance, EJ the Josephson energy,φ0 ¼ ℏ=2e the reduced flux quantum, Q the charge, andϕ ¼ R t−∞ dτVðτÞ the generalized flux. As usual, the chargeand the flux are conjugate quantum operators obey-ing ½ϕ; Q� ¼ i.Expanding the cosine and introducing bosonic annihi-

lation (creation) operator að†Þ, one obtains [44]

HJPA ¼ ω0a†a − EJ

X∞n>1

ð−Φ2ZPFÞn

ð2nÞ! ðaþ a†Þ2n; ð11Þ

with ω0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi8EJEC

pthe bare frequency of the resonator,

EC ¼ e2=2C the charging energy, and ΦZPF ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiEC=ω0

pthe unitless zero-point flux fluctuations.As JPAs are usually weakly nonlinear devices, the next

step is to perform a quartic potential approximation bykeeping only the first correction to the standard harmonicoscillator

HJPA ≈ ω0a†aþ Λ6ðaþ a†Þ4; ð12Þ

with the Kerr coefficient Λ ¼ −EJΦ4ZPF=4 ¼ −EC=2. As

the leading neglected correction is of order Λ2=ω0, this

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approximation is valid in the regime jΛj=ω0 ≪ 1, which isrelevant for typical JPA frequencies and Kerr nonlinearitiescorresponding to jΛj=ω0 ∼ 10−2 to 10−6 [44]. We note thatthe transmon qubit has the same circuit and Hamiltonian asa JPA but operates at larger Kerr nonlinearities [45].In the rotating-wave approximation (RWA), also valid

for jΛj ≪ ω0, we obtain the standard Kerr Hamiltonian

HKerr ¼ ~ω0a†aþ Λa†2a2; ð13Þ

with ~ω0 ¼ ω0 − 2Λ the renormalized oscillator frequency.By normal ordering and performing the RWA prior to thequartic potential approximation, corrections from all ordersrenormalize the resonator frequency and the Kerr non-linearity [46]. To keep the notation light, we neglect thissimple renormalization of parameters in the present work.While the Kerr Hamiltonian is obtained above from the

lumped-element circuit, the same Hamiltonian with renor-malized parameters applies for a distributed nonlinearresonator in the single-mode approximation [9,44] or alumped-element JPA with additional linear inductances[36]. However, it is worth noting that in both cases, theadditional inductances can reduce the validity of the quarticpotential approximation. See Ref. [9] for details.As we discuss in Sec. II, in order for this system to act

as a parametric amplifier, a pump must modulate one ofthe parameters at twice the resonance frequency. We nowconsider three pumping schemes leading to such amodulation.

B. Monochromatic current pump

We first consider the standard current-pumped JPA[1–4,7,9,33]. In this scheme, a single-current pump nearresonance with the oscillator is used. Because of theJosephson relations, the junction acts as a current-dependent nonlinear inductance with [47]

LNLðIÞ ¼ LJ

�1þ 1

6

�IIc

�2

þ � � ��: ð14Þ

For a monochromatic current pump Ip ∝ cosωpt, thefirst nonlinear contribution to the inductance is the cosinesquared, which leads to a modulation of the inductance attwice the pump frequency.Using standard circuit quantization techniques [48] and

using the quantum-optics language, the current pump isequivalent to adding a single-photon drive, such that thetotal Hamiltonian of the pumped circuit is

H1 ¼ HKerr þ ϵe−iωpta† þ ϵ�eiωpta; ð15Þ

where ϵ and ωp are the pump amplitude and frequency. In aframe rotating at the pump frequency, it is useful toeliminate the pump Hamiltonian using a displacementtransformation, which leads to a → αþ d, with α the

classical field and d the quantum fluctuations [33,39];see Appendix B for details. In this displaced frame, theHamiltonian takes the form

H01 ¼ Δ1d

†dþ λ12d†2 þ λ�1

2d2 þ H1c; ð16Þ

with the shifted detuning due to the cavity population

Δ1 ¼ ~ω0 þ 4jαj2Λ − ωp; ð17Þ

the effective parametric pump strength λ1 ¼ 2α2Λ, and theclassical field α obeying the nonlinear differential equation

i _α ¼ ϵþ�~ω0 − ωp þ 2Λjαj2 − i

κ

2

�α: ð18Þ

In general, the steady state of this cubic equation canexhibit bifurcation physics [33,47]. However, for thecurrent-pumped JPA, the bifurcation threshold coincideswith the parametric threshold, and in the context of para-metric amplification, parameters are chosen to be below thebifurcation point [39]. Thus, in what follows, αðtÞ is alwaysa single-valued function.For simplicity, H0

1 in Eq. (16) is obtained by performingthe quartic approximation before the displacement trans-formation. However, one can perform the displacementtransformation on the full cosine potential before perform-ing the RWA and the quartic potential approximation.These corrections, which mainly shift the operating fre-quency and bifurcation point of the amplifier, are studied indetail in Ref. [38].The displaced Hamiltonian of Eq. (16) also includes the

nonlinear corrections to the single-pump scheme

H1c ¼ μd†2dþ μ�d†d2 þ Λd†2d2; ð19Þ

with the cubic term coefficient μ ¼ 2αΛ and the standardquartic Kerr coefficient Λ. These corrections originatefrom the displacement of the Kerr nonlinearity. In orderto obtain linearized equations, they can be neglected in thesmall quantum-fluctuations limit [33]. In the followingsections, we explore the validity of this approximationand show that it is valid in the low-gain and low-Kerr-nonlinearity regime.When neglecting the corrections H1c, the displaced

Hamiltonian H01 can be related to HDPA with the mapping

Δ1 → Δ, λ1 → λ, d → a, and dout ¼ aout −ffiffiffiκ

pα → aout.

Thus, in this linear regime, the monochromatic current-pumped JPA is equivalent to the DPA but in a displacedframe. From the lab frame, this implies that while theoutput of a DPA is squeezed vacuum, the output of thiscurrent-pumped JPA is a displaced squeezed state.


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C. Bichromatic current pump

We now consider an alternative scheme with two currentpumps of frequencies ω1 and ω2 that are chosen such thatω1 þ ω2 ≈ 2 ~ω0 [34]. As shown in Fig. 2, while in themonochromatic case, the pump is at a frequency close tothe amplified signal [Fig. 2(a)], in the bichromatic case, thepumps are separated spectrally from the signal [Fig. 2(b)].Similar to Eq. (15), the starting Hamiltonian is

H2 ¼ HKerr þX2n¼1

½ϵne−iωnta† þ H:c:�; ð20Þ

with ω1;2 and ϵ1;2 the frequencies and amplitudes of the twopumps. In order to remove the pumps in a similar way asin the monochromatic case, we consider two displacementtransformations instead of one. This allows us to considertwo classical fields, each rotating at one of the pumpfrequencies, and to separate them from the quantumfluctuations of the cavity mode possibly rotating at a thirdfrequency.Choosing a frame rotating at the average pump fre-

quency Ω12 ¼ ðω1 þ ω2Þ=2 ≈ ~ω0, the double displacementtransformation leads to

a → de−iΩ12t þ α1e−iω1t þ α2e−iω2t: ð21Þ

A more detailed and formal treatment of the transformationis presented in Appendix B. By applying this transforma-tion, one obtains

H02 ¼ Δ2d

†dþ λ22d†2 þ λ�2

2d2 þ HR þ H2c; ð22Þ

with the shifted detuning

Δ2 ¼ ~ω0 þ 4Λðjα1j2 þ jα2j2Þ − Ω12; ð23Þ

and the effective parametric pump strength λ2 ¼ 4Λα1α2.All the rotating terms are grouped in the Hamiltonian

HR ¼ 8ΛRefα1α�2e−iΔ12tgd†dþ ½Λðα21e−iΔ12t þ α22e

iΔ12tÞd†2 þ H:c:�þ ½2Λðα1e−iΔ12t=2 þ α2eiΔ12t=2Þd†2dþ H:c:�; ð24Þ

where Δ12 ¼ ω1 − ω2 is the detuning between the twopump frequencies. Finally, we define

H2c ¼ Λd†2d2; ð25Þ

the nonlinear correction to the Hamiltonian.The rotating Hamiltonian HR can be dropped by a RWA.

Assuming symmetric pumps (α1 ∼ α2), this RWA is validfor Δ12 ≫ 8Λα1α2 ¼ 2λ2. Since the effective parametricpump strength is bounded by the parametric threshold λcritdefined in Eq. (9), one can choose the detuningΔ12 in orderto enforce the validity of the RWA. For zero detuning inEq. (22) (Δ2 ¼ 0), the RWA condition is simply Δ12 ≫ κ.An advantage of the bichromatic pump compared to themonochromatic pump is, thus, that the cubic terms(μd†2dþ H:c:) are now rotating and can be safelyneglected. The elimination of the cubic terms reducesthe nonidealities of the amplifier and implies that withrespect to the monochromatic current pump, the bichro-matic current-pumped JPA acts as an ideal phase-sensitiveamplifier for a larger parameter range (see Sec. IV).Again, in the small quantum-fluctuations limit, we can

neglect the higher-order correction H2c. Under this approxi-mation, the system is related to the DPA with the mappingΔ2 → Δ, λ2 → λ, d → a, and

dout ¼ aout −ffiffiffiκ

p ðα1e−iΔ12t=2 þ α2eiΔ12t=2Þ → aout: ð26Þ

Whereas in the monochromatic case, the pump leads to adisplacement of the output field at the center frequency of theamplified band, in this case, the two pumps lead to displace-ments far detuned from the band of interest. This spectralseparation allows us to filter the pumps without the need fora more involved pump-cancellation scheme [37].

D. Monochromatic flux pump

Current pumps are an indirect way to modulate theinductance of the JPA by using the nonlinearity of theJosephson junction. A well-studied alternative is to usean adjustable inductance that can be directly modulated.In superconducting circuits, this can be done by replacinga single Josephson junction with a superconductingquantum-interference device (SQUID), a flux-dependentnonlinear inductance. With this slight modification of thecircuit, parametric amplification can be obtained by fluxpumping. This pumping scheme has been extensivelystudied both experimentally [5,36,49] and theoretically[35]. Here, we derive the Hamiltonian of the flux-pumped

(a) (b)

FIG. 2. Relative position of the current-pump frequencies(black vertical arrows), and the frequency band of amplification(dark blue region) is shown for (a) the monochromatic currentpump with a single pump of frequency ωp ≈ ω0 and (b) thebichromatic current pumps with two pumps of frequenciesω1 þ ω2 ≈ 2ω0.


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JPA similar to Ref. [35], with an additional focus on higher-order corrections to the Hamiltonian.Replacing the Josephson junction by a SQUID modifies

the Josephson energy in the equations of Sec. III A suchthat [50]

EJ → EJ cos

�Φx

2φ0

�; ð27Þ

whereΦx is the external flux applied in the SQUID loop. Inorder to obtain parametric amplification, the external flux ismodulated at frequency ωp ≈ 2 ~ω0, with an additional staticcomponent chosen such that [35]

Φx

2φ0

¼ F þ δf cosωpt; ð28Þ

with F the unitless static flux and δf the modulationamplitude.In order to separate the harmonics of the pump, we

Fourier expand the flux-dependent Josephson energy

EJ cosðF þ δf cosωptÞ ¼Xn

EðnÞJ cosðnωptÞ; ð29Þ

where the coefficients of the expansion EðnÞJ are given in

Appendix C. In the relevant limit of small pump amplitude(δf ≪ 1), one obtains that the leading contribution of each

Fourier coefficient is EðnÞJ ∝ ðδf=2Þn=n!, and the expansion

of Eq. (29) can be safely truncated.Because of the flux modulation, the result of the RWA

on Eq. (12) is modified, and the Hamiltonian is

Hf ¼ Δfa†aþ λf2ða†2 þ a2Þ þ Hfc; ð30Þ

with the detuning Δf ¼ ~ω0 − ωp=2 and the effective

parametric pump strength λf ¼ Eð1ÞJ Φ2

ZPF=2. In the quarticpotential approximation, the nonquadratic corrections tothe Hamiltonian are now

Hfc ¼ Λfa†2a2 −Eð1ÞJ Φ4

ZPF

12ða†a3 þ a†3aÞ

−Eð2ÞJ Φ4

ZPF

48ða4 þ a†4Þ; ð31Þ

with Λf ¼ −Eð0ÞJ Φ4

ZPF=4 ¼ ΛJ0ðδfÞ cosF the Kerr non-linearity renormalized by the flux modulation.While the Kerr nonlinearity is essential for parametric

amplification in the current-pumped cases, the parametricpump strength of the flux-pumped JPA is independent ofthis quantity. It is merely an artifact of the use of Josephsonjunctions to build a flux-dependent inductance. Moreexplicitly, in the limits of a resonant pump Δf ∼ 0 and a

small pump amplitude δf ≪ 1, the ratio of the parametricpump strength to the parametric threshold is

jλf=λcritj ≈ δfQ tanF; ð32Þ

with Q ¼ ω0=κ the JPA quality factor [36]. This ratio isindependent of the Josephson energy or of the Kerrnonlinearity.For standard JPA quality factors Q ∼ 10–100 and static

flux bias such that tanF ≳ 1, Eq. (32) implies that para-metric pump strengths close to the parametric thresholdcan be obtained even in the small flux-pump limit δf ≪ 1.Thus, the leading correction to the JPA Hamiltonian is thefirst term of Eq. (31), which is independent of δf. The othercorrections, respectively, linear and quadratic in δf, can bedropped in this small flux modulation limit. This impliesthat the leading higher-order correction to the flux-pumpedJPA is the same as in the bichromatic current-pump case(Hfc ≈ H2c), and, thus, the flux-pumped JPA Hamiltonianreads

Hf ≈ HDPA þ H2c: ð33Þ

Again, in the limit of small quantum fluctuations, higher-order corrections can be dropped, and the JPA related tothe DPA model with the mapping Δf → Δ and λf → λ.Since there is no displacement transformation, this pump-ing scheme is more closely related to the DPA than thecurrent-pumped JPAs.

E. Summary and comparison of pumping schemes

Table I compares and summarizes the pumping schemesreviewed in this section. It is divided into two parts: thefirst presents the general properties of each pumpingscheme, while the second summarizes the expressionsfor the parameters and leading higher-order correctionsto the ideal DPA Hamiltonian.The first distinction to make between these schemes is the

spatial and spectral separation of the pump and signal. In theflux-pump case, two distinct ports are used for the signal andthe pump, while in the current-pump cases, the same inputport is used for both the pump and the signal. Hence, whilethe output of the flux-pumped JPA is squeezed vacuum, theoutput of the current-pumped JPAs is displaced due toreflected pump field(s). In the bichromatic case, these fieldsare far detuned from the amplified signal and can be filteredout prior to the measurement. On the contrary, in themonochromatic case, the reflected pump is at the frequencywhere the amplifier gain is maximal. This implies that greatcare must be taken to either cancel the pump or separate itfrom the signal [37].Moreover, all three pumping schemes lead to a negative

pump-induced frequency shift of the resonator. While inthe case of the current pumps, the shift follows from thepopulation of the nonlinear cavity by the pump field(s), in


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the case of the flux pump, the shift is a geometric effect dueto the cosinusoidal dependence of the SQUID’s Josephsonenergy on the external magnetic flux [35,49]. These pump-induced detunings obey the relation

jΔðpÞf j ≪ jΔðpÞ

1 j ≤ jΔðpÞ2 j; ð34Þ

where the ðpÞ superscript is used to note that we areconsidering the pump-induced shift. Note that for the currentpumps, this shift scales asQ−1, while in the flux-pump case,it scales as Q−2 (see Table I).Finally, we note that while the leading higher-order

correction is the same for the bichromatic current pumpand the flux pump, the monochromatic current-pumpHamiltonian has additional cubic corrections. This impliesthat without any change to the actual amplifier circuit (fixedparameters), using a bichromatic current pump instead of amonochromatic current pump decreases the nonidealitiesof the JPA.

IV. SIGNATURE OF HIGHER-ORDERCORRECTIONS IN THE INTRACAVITY FIELD

In order to evaluate the impact of the higher-ordercorrections that we discuss in the previous section, wenumerically compute first- and second-order moments ofthe steady-state intracavity field. These quantities show thatwhile the JPA acts as an ideal DPA in the low nonlinearityregime, the higher-order corrections can play a significantrole in the presence of larger nonlinearities. To give a moreintuitive representation of the effect of the nonidealities onthe amplifier state, we also calculate Wigner functions ofthe intracavity field.The numerical results of this section are obtained by

finding the steady-state solution of the master equation

_ρ ¼ −i½HDPA þ Hα; ρ� þ κD½d�ρ; ð35Þ

where Hα ¼ H1c, H2c are the higher-order correctionsconsidered, and D½d�ρ ¼ d ρ d† − ðd†d ρþρd†dÞ=2 is thestandard dissipation superoperator [26,27].

A. Deviation from standard DPA results

With the ideal DPA model, the first-order moment of theintracavity field is hdi ¼ 0. However, in the case of themonochromatic current-pumped JPA, the cubic correctionsact as an effective pump leading to an additional displace-ment of the field and, thus, a nonzero first moment.To understand this effect, one can consider a mean-fieldtreatment of the cubic corrections, where [9]

μd†2dþ H:c: ≈ ½ð2μnþ μ�mÞd† þ H:c:�; ð36Þ

with n ¼ hd†di, and m ¼ hd2i the second-order momentsof the JPA state. Under this approximation, the cubicterms act as an additional state-dependent pump and canbe eliminated by a second displacement transformation.Such a mean-field treatment was used to study pumpdepletion effects and the dynamic range of the JPAin Ref. [9].A mean-field solution requires the self-consistent sol-

ution for n, m. Instead, we numerically find the steady stateof the master equation (35). Figure 3(a) shows the ratio ofthe displacement induced by the cubic terms to the steady-state solution of Eq. (18) for the displacement field α.While negligible in the low nonlinearity limit, the induceddisplacement becomes significant for larger Kerr non-linearities. On the contrary, and as expected from mean-field theory, no additional displacement is observed for thebichromatic current and monochromatic flux-pump cases.We now consider signatures of the corrections in the

second-order centered moments Mi ¼ hd2i − hdi2 andNi ¼hd†di− jhdij2, where the subscript i¼DPA, JPArefers to the model that is used. It follows from

TABLE I. Comparison of pumping scheme properties, parameters, and leading Hamiltonian corrections.

Pumping scheme Monochromatic current Bichromatic current Monochromatic flux

Spectral separation No Yes YesSpatial separation No No YesOutput statea DðαÞSðξÞj0i Dðα1eiΔ12t þ α2e−iΔ12tÞSðξÞj0i SðξÞj0iEffective parametric pump (λ) 2Λα2 4Λα1α2 4ΛfðEð1Þ

J =ω0ÞPump-induced frequency shift (ΔðpÞ

1;2;f) 4Λjαj2 4Λðjα1j2 þ jα2j2Þ ω0

� ffiffiffiffiffiffiffiffiffiffiffiffiffiJ0ðδfÞ

p− 1�

Relative shift (ΔðpÞ1;2;f=ω0)

b −jλ1=λcritj=Q −jλ2=λcritjðxþ x−1Þ=2Q −jλf=λcritj2=2 ~Q2

Corrections (Hð1;2;fÞc) μd†2dþ μ�d†d2 þ Λd†2d2 Λd†2d2 Λfa†2a2

aWe use the standard notation of quantum optics, where DðβÞ and SðξÞ are, respectively, the displacement and squeezingoperators [51].

bWe introduce the compact notation x ¼ jα1=α2j, and ~Q ¼ Q tanF.


054030-7

Heisenberg uncertainty principle, that for any state, thesemoments obey the relation

jMij ≤ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNiðNi þ 1Þ

p; ð37Þ

where the equality is obtained only for pure states [27].From the analytical solution to the DPA model (see Sec. IIand Appendix A), one obtains a value of jMDPAj below thisbound

jMDPAj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNDPAðNDPA þ 1=2Þ

p: ð38Þ

This result can be understood from the fact that due todamping, the steady state of the DPA is not a pure squeezedstate [42].To quantify the deviation of the JPA moments from the

expected results of an ideal DPA due to the corrections, wedefine the deviation

ΞJPA ¼ 1 −jMJPAjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

NJPAðNJPA þ 1=2Þp ; ð39Þ

where MJPA, NJPA are the centered moments of the JPAintracavity field, here calculated numerically includinghigher-order corrections. The deviation ΞJPA is zero in thecase of an amplifier that maps exactly to a DPA (negligiblehigher-order corrections) and increases towards 1 as thecorrections to the DPA Hamiltonian become important.

As observed in Fig. 3(b), the deviations increase withthe amplitude of the Kerr nonlinearity. The additionalcubic terms of H1c lead to larger deviation for the sameparameters. Thus, by simply using two current pumpsinstead of one, the deviation from the expected results for aDPA is reduced by approximately 2 orders of magnitude.To put in context the range of Kerr nonlinearity considered,the vertical lines indicate the approximate experimentalparameters of three recent experiments with JPAs [7,29,37].We note that in practice, the smaller nonlinearity (dashedgray line) is obtained using junction arrays. Indeed, theKerr nonlinearity with a junction array is inversely propor-tional to the square of the number of junctions in thearray [4,9].

B. Phase-space representation

In order to visually represent the deviation from theDPA, we present in Fig. 4 the Wigner function of theintracavity field for increasing Kerr nonlinearities (from leftto right). For each value of jΛj=κ, we present results forthe monochromatic current-pumped JPA (including cubicand quartic corrections H1c) in the top panels and for thebichromatic current-pumped (quartic correction H2c) or,equivalently, the flux-pumped JPA in the bottom panels.In all cases, we consider a phase-preserving gain ofG ¼ 16 dB. While in the low nonlinearity case presentedin Fig. 4(a), we observe a quadrature squeezed state forboth types of corrections, and in the increased nonlinearityof Fig. 4(b), non-Gaussian signatures appear in the mono-chromatic pump case. Increasing even more the nonlinear-ity amplitude in Figs. 4(c) and 4(d), both types ofcorrections lead to non-Gaussian signatures.In the limit of large nonlinearity jΛj=κ, we observe that

the shape of the Wigner function varies with the pumpingscheme. In the monochromatic current-pump case, weobserve the so-called crescent- or banana-shaped defor-mation of the distribution [52]. Similar distributions areobserved in the transient dynamics of a driven Kerr non-linear resonator. In particular, the large nonlinearity regimejΛj ≫ κ has been well studied both theoretically [53] andexperimentally in superconducting circuits [54].In the case of a single Kerr-type correction, we observe a

more symmetric “S”-shaped Wigner function deformation.In the case of the flux-pumped JPA, S-shaped features havebeen predicted theoretically and observed experimentallyusing a semiclassical analysis of the phase dependence ofthe JPA response [30,35]. This deformation of the Wignerfunction is also consistent with experimental studiesimaging the JPA output field; see Sec. VI D.From the observed deformation of the Wigner function

in Figs. 4(c) and 4(d), one can expect the higher-ordercorrections to limit the squeezing produced by a JPA. Sucha saturation of squeezing has been observed experimentally[28,29] and is discussed in more detail in Sec. VI B. Inaddition, one can expect the output field of the JPA to

(a)

(b)

FIG. 3. (a) Ratio of the displacement induced by higher-ordercorrections to the steady-state solution of Eq. (18). The Kerrcorrection (bichromatic current pump and monochromatic fluxpump) induces no displacement. (b) Deviation from standardDPA results as defined in Eq. (39). From left to right, the verticallines are approximate Kerr nonlinearity for the experiments ofRef. [37], Ref. [7], and Ref. [29], respectively. The amplitudeof the Kerr coefficient Λ is varied for a fixed gain G ¼ 16 dB(λ1;2 ¼ 0.85λcrit) with Δ1;2, γ ¼ 0.


054030-8

exhibit significant non-Gaussian signatures for large gainand nonlinearity. Both the squeezing level and the non-Gaussian character of the output field are characterizedmore quantitatively in Sec. VI. More generally, the resultsof this section indicate that for the same parameters,the additional cubic terms in the monochromatic current-pump case lead to additional nonidealities limiting JPAperformance.

V. GAIN AND QUANTUM EFFICIENCY

To characterize the effect of higher-order corrections onamplifier properties, in this section, we numerically com-pute the low power gain and added-noise number of theJPA. From these, we obtain the phase-sensitive and phase-preserving quantum efficiency of the amplifier.

A. Phase-sensitive and phase-preserving gain

The gain is computed by considering the linear responseof the JPA to a narrow-band signal probe or, in other words,the same approach as is used experimentally. More con-cretely, starting from the steady-state solution of the JPAmaster equation (35), we add to the Hamiltonian the probefield

Hprobe ¼ ϵprobea† þ ϵ�probea; ð40Þ

with ϵprobe the drive amplitude and find the new steady stateunder this weak drive. From the displacement of the cavityfield generated by the probe, one can calculate the outputfield response. As already clear from Eq. (5), the gain is afunction of the probe frequency. Using a time-independentprobe, we compute the JPA gain at the center frequencyω ¼ 0.As we discuss in Sec. IV, the cubic corrections in the

Hamiltonian of the monochromatic current-pumped JPAalso induce a displacement of the cavity field. This addi-tional displacement modifies the bifurcation point of the

system, something that can lead to instabilities in thenumerical calculations. Hence, this section considers onlythe effect of a Kerr-type correction, which is the leadingcorrection for the bichromatic current pump and themonochromatic flux pump. In the case of the monochro-matic current pump, the additional cubic terms can lead tocorrections which, following the results of the previoussections, will appear at lower gain and nonlinearity thanthose due to the quartic term.To study the linear response regime, we limit our

analysis to a low-power probe where ϵprobe ≪ κ. Whilean analysis of the dependence of the gain on the probeamplitude would allow us to calculate the dynamic range ofthe JPA [9,38], this is beyond the scope of this paper.Using the input-output relation Eq. (4), the displacement

generated by the probe and the input field amplitudehaini ¼ iϵprobe=

ffiffiffiκ

p, we can compute the gain matrix g,

whose elements are defined by the linear input-outputequation

�Xout

Pout

�¼�g11 g12g21 g22

��Xin

Pin

�; ð41Þ

where we define the standard quadratures X¼ðaþ a†Þ= ffiffiffi2

pand P ¼ iða† − aÞ= ffiffiffi

2p

. By considering separately theresponse to two probes with orthogonal phases, we cancalculate all elements of the gain matrix.To obtain the phase-preserving gain, we express the

above quadrature input-output relation in terms of fieldoperators. Using Eqs. (5) and (8), we then find that the JPAphase-preserving gain is related to the phase-sensitive gainmatrix by

~G ¼ 1

4jg11 þ g22 þ iðg21 − g12Þj2: ð42Þ

Figure 5 presents the phase-preserving gain, as well asthe elements of the phase-sensitive gain matrix for a JPA

(a) (b) (c) (d) FIG. 4. Steady-state Wi-gner function WðβÞ of theJPA intracavity field for again G ¼ 16 dB (λ1;2;f¼−0.85λcrit) with Δ1;2;f, γ¼0. Numerical result ob-tained via master equationsimulation of the Hamilto-nians of the monochroma-tic current pump Eq. (16) inthe displaced frame (toprow) and of the bichromaticcurrent pump Eq. (22) afterRWA, which is equivalentto the monochromatic fluxpump (bottom row).


054030-9

with increasing Kerr nonlinearities. These quantities areplotted as a function of the phase-preserving gain calcu-lated from Eq. (8) for an ideal DPA. As expected, Fig. 5(a)shows that in the low nonlinearity regime (blue diamonds),the JPA phase-preserving gain is equal to the ideal DPAgain, and the corrections are negligible. As the Kerrnonlinearity increases (green squares and red circles), theJPA nonidealities result in a decreased gain, with deviationsincreasing with gain.Figure 5(b) presents the diagonal elements of the phase-

sensitive gain matrix, while Fig. 5(c) shows the off-diagonal elements. In the low-gain regime, as expectedfor a phase-sensitive amplifier, the diagonal elements areinversely proportional with one quadrature amplified andthe other attenuated. In this regime, the gain matrix isdiagonal. When the off-diagonal terms become significant,the attenuation coefficient g22 starts to increase, deviatingsignificantly from the expected behavior of an ideal phase-sensitive amplifier.For higher gain and nonlinearities, the gain matrix is not

symmetric and cannot be diagonalized with orthogonaleigenvectors. In this regime, the amplification processmixes quadratures. Noting that a similar effect occurs fora DPAwith pump-cavity detuning [39], we can interpret the

effect of the nonlinearity as a gain-dependent detuning ofthe system. Hence, for a given gain, choosing a slightlydifferent detuning can mitigate higher-order effects andreduce quadrature mixing. This implies that when higher-order corrections are important, the optimal phase andfrequency of operation of the JPA is gain dependent. Theinterplay of the optimal frequency of operation and non-linear corrections has recently been the subject of exper-imental investigation in a similar device [55].

B. Phase-preserving quantum efficiency

To characterize the effect of Kerr-type correction on theJPA noise properties, we now evaluate its added noise andquantum efficiency. As we illustrate in Fig. 6, the quantumefficiency η can be interpreted as the transparency of afictitious beam splitter added at the input of a noiselessamplifier in order to model the noise added by theamplification as additional input vacuum noise [15,56].With this picture in mind, we define the quantum

efficiency η such that the input-output field fluctuationsare related by

hjaoutj2i ¼Gη

�ð1 − ηÞ 1

2þ ηhjainj2i

�; ð43Þ

with hj · ji the symmetrized fluctuations of an operator

hjOj2i ¼ 1

2hO†Oþ OO†i: ð44Þ

Note that the case η ¼ 0 corresponds to having no outputsignal and is, therefore, not relevant here. The first term ofEq. (43) is the added noise due to the amplification processrepresented as vacuum fluctuations in Fig. 6, while thesecond term is fluctuations in the input signal. It is useful toexpress Eq. (43) in a simpler form

hjaoutj2i ¼ GðAþ hjainj2iÞ; ð45Þ

where A ¼ ð1 − ηÞ=2η is the added noise referred to theinput. Using the inequality hjOj2i ≥ h½O; O†�ij=2, onecan derive the well-known quantum limit for a phase-preserving amplifier [24]

(a)

(b)

(c)

FIG. 5. (a) Numerical phase-preserving gain for a JPA withKerr-type correction H2c calculated from Eq. (42) versus gainfor the DPA calculated from Eq. (8). (b) Norm of the diagonalelements g11 (solid curves) and g22 (dashed curves) of the phase-sensitive gain matrix defined at Eq. (41). (c) Off-diagonalelements jg21j2 (solid curves) and jg12j2 (dashed curves). Forall curves, the gain is calculated for ω ¼ 0, Δ ¼ 0, and γ ¼ 0.

FIG. 6. Schematics of the quantum-efficiency definition as abeam splitter with transparency η and input vacuum noise. Theblue disks represent signal noise before and after amplification,while red disks represent noise added by the amplificationprocess.


054030-10

A ≥1

2

�1 −

1

G

�; ð46Þ

which simplifies in the large gain limit to A ≥ 1=2.Using these definitions, the quantum efficiency can be

expressed as

η ¼ 1

1þ 2A≤

G2G − 1

; ð47Þ

where we use the quantum limit of Eq. (46). This inequalityimplies, in the large-gain limit, that the bound on thequantum efficiency of a phase-preserving measurement isη ≤ 1

2. This result simply reflects the well-known fact that

ideal phase-preserving amplification can be obtained by theuse of two ideal phase-sensitive amplifiers and a beamsplitter which adds vacuum fluctuations [24].Note that the alternative definition η ¼ 1=ð1þ NÞ ¼ 2η

of the quantum efficiency is also found in the literature,with N ¼ A − 1=2 a number of added-noise photons. Thisdefinition relates the amplifier performance to an idealphase-preserving amplifier instead of a noiseless amplifier.However, this expression implicitly assumes the large-gainlimit of Eq. (46) and, therefore, overestimates the quantumefficiency in the low-gain regime where 1=G cannot beneglected. In this work, we consider the definition ofEq. (47), as this allows us to treat phase-preserving andphase-sensitive amplification on the same footing and isindependent of gain.Figure 7 shows the quantum efficiency in the phase-

preserving case as a function of gain and for three Kerrnonlinearities. In the low-gain regime, all curves are equalto the quantum limit. For higher gains, the Kerr nonlinearityleads to a decreased quantum efficiency. These results areobtained by numerically calculating the gain and the outputfield spectrum of the amplifier, with

hjaoutj2i ¼ nout½0� þ1

2: ð48Þ

In this calculation, we neglect any bandwidth or detuningeffect and consider the zero-frequency component of thespectrum.Surprisingly, even for a significant phase-preserving gain

~G ¼ 20 dB and Kerr nonlinearity jΛj ¼ 0.01κ, the JPAwith Kerr-type corrections is nearly quantum limited with-out any tuning of parameters. On the contrary, we show inthe following section that the same Kerr-type correctionstrongly influences the quantum efficiency of a phase-dependent measurement. In that case, a careful choice ofthe phase of operation of the JPA is essential in order toobtain near-quantum-limited amplification.

C. Phase-sensitive quantum efficiency

We now generalize the concepts of the previous sectionto the case of phase-sensitive amplification. Contrary tothe simpler case of phase-preserving amplification, thequantum efficiency is not a single number but rather afunction of the measurement phase θ. Hence, one mustchoose the measured quadrature in order to maximizequantum efficiency.For the ideal DPA, no noise is added independently of

the phase considered and ηðθÞ ¼ 1 for all θ. In that specificcase, the quantum efficiency of the full measurement chain[57] is maximized for the measurement phase θm whichmaximizes the gain g11. However, when including a Kerrcorrection, we show in Sec. VA that the gain matrixbecomes nonsymmetric. In that case, the nonzero g12 leadsto quadrature mixing during the amplification, which canbe seen as a source of added noise. Thus, we show thatcontrary to the ideal DPA, in the presence of a Kerrcorrection, the maximal quantum efficiency is not obtainedby maximizing g11 but rather by minimizing g12. We noteθ0 the phase of the quadrature which minimizes the mixingof noise with the signal.More formally, in order to characterize the field fluctua-

tions, we define the matrix of the symmetrized moments asthe matrix analog of Eq. (44) [24]

σj ¼ hΔX2

ji 12hfΔXj;ΔPjgi

12hfΔXj;ΔPjgi hΔP2

ji

!; ð49Þ

with j ¼ out, in, and fA; Bg ¼ A BþB A the anticommu-tator. Similarly, the added-noise matrix is defined through ageneralization of Eq. (45)

σout ¼ gðσA þ σinÞgT: ð50Þ

We note that the product of the diagonal elements ofthis matrix are bounded by the quantum limit to amplifi-cation [24]

FIG. 7. Phase-preserving quantum efficiency (η) as a functionof gain for a JPA with Kerr correction. The two lower Kerrnonlinearity cases considered jΛj=κ ¼ 10−4 (blue diamonds) andjΛj=κ ¼ 10−3 (green squares) are near quantum limited and, as aresult, are difficult to resolve from the quantum limit (dashedblack curve) for the full range of parameters considered. For allcurves, the quantum efficiency is calculated for ω ¼ 0, Δ ¼ 0,and γ ¼ 0.


054030-11

σA11σA22 ≥1

4

1 − 1

g11g22

: ð51Þ

For the standard lossless DPA, we obtain noiseless ampli-fication since g22 ¼ 1=g11 and σA11 ¼ σA22 ¼ 0.In the presence of Kerr correction, to make explicit the

choice of the measurement phase, we define the phase-dependent added noise

AðθÞ ¼ ½RTðθÞσARðθÞ�11 ð52Þ

as the first diagonal element of the rotated added-noisematrix, with RðθÞ the counterclockwise orthogonal rotationmatrix. From Eq. (52), we define the phase-dependentquantum efficiency ηðθÞ ¼ 1=½1þ 2AðθÞ� ≤ 1. While wefocus in the following on the diagonal element of theadded-noise matrix, with AðθÞ characterizing the noiseadded to the amplified quadrature, in general, σA isnondiagonal, and the amplification can lead to addedcross-correlations between the quadratures.Figure 8 shows the quantum efficiency and gain as a

function of the quadrature phase θ for increasing values ofjΛ=κj. In Fig. 8(a), we observe that the quantum efficiencyoscillates with θ and becomes increasingly peaked around avalue of the phase with increasing nonlinearity. For a fixednonlinearity, increasing the gain leads to the same effect(not shown). We note that the position of the peak in thequantum efficiency correlates with a dip in the off-diagonalmatrix element g12 shown in Fig. 8(b). This dip is shifted toa narrower range of phases as the Kerr nonlinearity is

increased. This is in agreement with Fig. 5(c), which showsthat for a fixed quadrature phase, increasing the non-linearity leads to larger g12 and, thus, requires a largerphase correction. Thus, as expected, the quantum efficiencyis maximized when the noise added through quadraturemixing is minimized.Figure 9 presents 1 − ηðθÞ as a function of phase-

sensitive gain for increasing Kerr nonlinearities. To furtherillustrate the importance of choosing the appropriate phase,results for both the optimal phase θo of minimal cross gainjg12j (dashed curves) and for the nonoptimal phase θm ofmaximal gain (jg11j, solid curves) are shown. As expected,the quantum efficiency is reduced [larger 1 − ηðθÞ] forincreased gain and nonlinearities. Strikingly, when consi-dering a gain of 25 dB and jΛ=κj ¼ 10−2, the quantumefficiency at phase θo is around 0.9, while it is close to zerofor θm. Figure 9(b) shows that this dramatic difference inperformance is obtained with less than 0.2-rad (12°)difference in phase.These results point to both the Kerr nonlinearity and the

nonoptimal choice of measurement phase as a possibleexplanation for a series of experiments that have reportedsmaller than expected quantum efficiencies for JPAs[17,29,31,32]. A more detailed experimental study of thequantum efficiency as a function of both detuning andmeasurement phase could confirm these results and wouldallow for a better understanding of these nonlinear effects.

VI. OUTPUT FIELD CHARACTERIZATION

Following the results of the previous sections, one canexpect to also find signatures of the nonidealities in the JPA

(a)

(b)

FIG. 8. (a) Phase-sensitive quantum efficiency as a functionof the phase of the measurement quadrature. (b) Matrix elementsof g (g11 solid curve, g12 dashed curve) as a function of thequadrature phase θ. All data correspond to a photon-numbergain GJPA ¼ 23 dB for a JPA with Kerr correction andω ¼ Δ ¼ γ ¼ 0.

(a)

(b)

FIG. 9. (a) Reduction in the phase-sensitive quantum efficiencydue to Kerr-type corrections. Solid curves correspond to thephase of maximal gain θm, while dashed curves correspond to thephase θo, where the cross gain g12 is minimal. (b) Shift in rad ofthe phases θm and θo as a function of gain.


054030-12

output field, including in squeezing experiments. In thissection, we compute moments of the output field whichallow us to characterize the JPA as a source of squeezedlight. Following the results of Sec. IV, and more generallyfor Kerr cavities [52,54], we expect the higher-ordercorrections to lead to a non-Gaussian output field. Toverify this, we calculate third- and fourth-order cumulantswhich reveal departure from Gaussianity.Throughout this section, we compare our numerical

results to an experimental characterization of the outputfield of a flux-driven JPA. Moments of the JPA output fieldare obtained using a single-path reconstruction method[58], while a full output field imaging is obtained from adeconvolution technique. Details of the experimental setupand methods are presented in Appendix D.

A. Filtered output field:Definition and numerical technique

In order to compare our numerical results to experi-mental data, we consider the finite bandwidth of themeasurement chain in our calculations. To this end, wedefine the filtered output field DðtÞ as the convolution ofthe full output field aout [59],

DðtÞ ¼ f⋆aoutðtÞ ¼Z

∞

−∞dτfðt − τÞaoutðτÞ; ð53Þ

with a filter function f normalized such thatR∞−∞ dtjfðtÞj2 ¼

1 in order to ensure standard bosonic commutation relations½D; D†� ¼ 1. The moments of this field can be evaluatedby numerical integration of correlation functions using thequantum regression formula [27]. The details of the numeri-cal technique are presented in Appendix E. We note that thecomplexity of the calculation scales exponentially with theorder of the moment considered [59]. As a result, fourth-order moments are at the limit of our current computationalcapacities. Fortunately, this is sufficient to characterize thedeparture from ideal Gaussian behavior. Unless otherwisespecified, the filter used is a time-domain boxcar filterof 256 ns (bandwidth of approximately 4 MHz), whichcoincides with the experimental method.

B. Squeezing level

In order to characterize the squeezing produced by theJPA, we calculate the filtered output field squeezing leveldefined as [28]

Sf ¼ hΔX2vaci

hΔX2mini

; ð54Þ

with hΔX2vaci ¼ 1=2 the variance of the vacuum state, and

hΔX2mini ¼ Nf þ

1

2− jMfj; ð55Þ

the minimal variance of the filtered field. Here, we definethe centered moments of the filtered output field Nf ¼hD†Di − jhDij2 and Mf ¼ hD2i − hDi2.Figure 10(a) shows the filtered squeezing level Sf as a

function of phase-preserving gain for a JPA with Kerrcorrection. For an ideal JPA, the squeezing level of thecenter frequency (Dirac-δ filtering) increases with gainwithout bound (dashed black curve). On the other hand,even for an ideal JPA (no higher-order corrections), thefiltered squeezing level saturates for a finite-bandwidthfilter (dotted-dashed gray curve). Indeed, as the gainincreases, the squeezing bandwidth is reduced and even-tually becomes smaller than the filter bandwidth. At thatpoint, nonsqueezed radiation contributes to Sf limiting themeasured squeezing level. This relation is a differentillustration of the gain-bandwidth trade-off in cavity-basedparametric amplifiers [60]. A more detailed discussion ofthis effect is given in Appendix F. The three remainingcurves show the effect of the Kerr nonlinearity on Sf as afunction of the numerically calculated gain [vertical axis ofFig. 5(a)]. As expected, while at low gain all curvesoverlap, for increasing gain, Sf reaches a maximal value,which decreases with Kerr nonlinearity.Figure 10(b) compares the squeezing level of a mono-

chromatic current-pumped JPA (up and down triangles) and

(a)

(b)

FIG. 10. (a) Squeezing level of the filtered output field (Sf)versus numerical calculation of the gain using Eq. (42). The filter isa time-domain boxcar filter of length 256 ns. Dashed black curve(dotted-dashed gray curve) is themaximal squeezing level of aDPAwithout (with) the filter. Solid curves are numerical results includ-ing H2c ≈ Hfc corrections. (b) Comparison of the squeezing levelwith H1c corrections (dashed curves) or H2c ≈ Hfc corrections(solid curves) as a function of the photon-number gain calculatedwithout corrections [Eq. (8)]. (All panels: κ=2π ¼ 50 MHz,Δ ¼ γ ¼ 0.)


054030-13

a bichromatic current-pumped JPA or monochromatic flux-pumped JPA (circle and squares) for two Kerr nonlinear-ities. As we discuss in Sec. V, the effect of nonidealities onthe gain could not be obtained in the case of the mono-chromatic current-pumped JPA. To compare pumpingschemes on equal footing, squeezing levels are shown asa function of the phase-preserving photon-number gaincalculated for an ideal DPA using Eq. (8). Note that thesolid curves for the bichromatic current-pumped JPApresent the same squeezing levels as in Fig. 10(a) but asa function of the ideal DPA gain. The curves have similarshapes for both pumping schemes. However, due to theadditional cubic corrections, the monochromatic current-pumped JPA achieves a smaller maximal squeezing levelthan is possible for a bichromatic current-pumped JPAwithsingle Kerr-type correction. Hence, for the same JPA, goingfrom a monochromatic to a bichromatic current pump leadsto a significant increase in the maximal squeezing level thatcan be produced. These results are in agreement with whatcan be expected from intracavity signatures in Sec. IV,where both the second-order moments and the Wignerfunctions present stronger nonidealities in the monochro-matic current-pump case.In order to confirm that these higher-order correction

effects explain the experimentally observed reduction insqueezing levels of JPAs [14,15,28,29], we compare ournumerical results to experimental data. Using a moment-based reconstruction method, the squeezing level of a flux-pumped JPA is measured [58,59,61]. Figure 11 showsexperimental results and numerical data together. Thereconstruction technique being highly sensitive to the gainof the measurement chain, the green diamonds (red squares)are the higher (lower) estimate of the squeezing level basedon the corresponding measurement chain gain estimate,while the error bars correspond to statistical error evaluation.The blue circles are numerical results where all parameters ofthe simulations are obtained from independent experiments

with no fitting parameters. As a comparison, the dotted-dashed light blue line indicates the result expected for anideal DPA. While there is quantitative agreement betweennumerics and experiment for the maximal squeezing levelmeasured, the overall agreement is only qualitative. Thesmall discrepancies could be due to spatial variations in theimpedance of the JPA environment that leads to an exper-imentally observed variation in the gain-bandwidth productof the JPA with increasing gain.We note that contrary to previous hypothesis [28], our

results indicate that the squeezing saturation happens atpump powers below the bifurcation threshold of the JPA.Our results strongly indicate the JPA higher-order correc-tion as the main contributing factor to the experimentallyobserved decrease of squeezing in the large-gain limit.

C. Cumulants

The Wigner functions of Fig. 4 clearly indicate that thehigher-order corrections lead to non-Gaussian intracavityfields. In order to characterize the non-Gaussianity of thefiltered output field, we compute third- and fourth-ordercumulants ⟪D3⟫ and ⟪D4⟫. In the case of a univariatedistribution, the third- (fourth-) order cumulant can benormalized to define the skew (kurtosis) of the distribution.While such definitions are not as straightforward in thecase of the multivariate distribution considered here,the third- and fourth-order cumulants still characterize thenon-Gaussian character of the field. Indeed, recall that acumulant of order n is a polynomial of moments of order nand less, and that for a Gaussian field, only the cumulantsof order one and two are nonzero [62].Figure 12(a) shows numerical calculation of j⟪D3⟫j for

the JPAwith a monochromatic current pump. The cumulantincreases following a power law with gain and nonlinearity.The numerical results for the bichromatic current andflux pump are not shown here, as they are exactly zero.Figure 12(b) shows a fourth-order cumulant for both typesof corrections. Again, in agreement with the results ofFig. 4, non-Gaussianity increases with gain and nonlinear-ity. In addition, we observe that the slope is larger forthe monochromatic current pump (cubic corrections). Theother third- and fourth-order cumulants ⟪D†D2⟫, ⟪D†D3⟫,and ⟪D†2D2⟫ are also calculated, and similar trends areobserved (data not shown).These numerical results present higher-order corrections

to the DPA Hamiltonian as a significant source of non-Gaussianity in the output field. In addition to the exper-imental data presented below, these corrections couldexplain previously reported experimental observation ofnon-Gaussian features for JPAs [15,28].

D. Output field imaging

In order to further investigate the non-Gaussian featuresthat we discuss above, we experimentally image the Husimi

FIG. 11. Experimental characterization (green triangles and redsquares) of the squeezing level of a monochromatic flux-driven JPA with comparison to numerical results (blue circles).Numerical calculations are performed with the parametersΛ=2π ¼ −1.55 MHz, κ=2π ¼ 130 MHz, and γ ¼ κ=10 obtainedfrom independent measurements (no fitting parameters). Theexperimental parameters, setup, and method are presented inAppendix D.


054030-14

Q function of the output field of a flux-driven JPA. Adeconvolution technique is used to extract the JPA outputfield from the noisy histograms resulting from sampling themeasurement chain output via homodyne detection. Thismethodology not only provides an experimental analysisof the higher-order cumulants of the output field but alsoprovides a direct image of the output field that can becompared with qualitative expectations based on the intra-cavity fields calculated in Fig. 4. See Appendix D forexperimental details of the method.Figure 13 presents a comparison of the output field for

low JPA gain (5 dB, top row) and high JPA gain (24 dB,bottom row). In the figures, both the raw noisy histo-grams, which are the convolution of the signal with noisedue to the amplification chain (left column), and theHusimi Q function extracted through the use of deconvo-lutions (right column) are shown. In addition, Fig. 13(e)presents the magnitude of the cumulants up to fourthorder extracted from the deconvolved distributions ofFigs. 13(b) and 13(d). As expected from the numericalcalculations, the low-gainQ function appears Gaussian upto experimental resolution, and third- and fourth-ordercumulants are small. On the contrary, the Q function athigh gain is non-Gaussian with a noticeable S-shapeddistortion, consistent with expectations based on theintracavity fields presented in Fig. 4 and also consistentwith a recent semiclassical analysis of the JPA response[30]. The inferred third- and fourth-order output field

cumulants presented in Fig. 13(e) clearly deviate from theideal value of zero expected for a Gaussian distribution.However, from the numerical results of Fig. 12 and thesymmetry with respect to the centroid of the Q functionin Fig. 13(d), one expects the role of the third-ordercumulants to be small. To this end, we note that the third-order cumulants are smaller than the second-order cumu-lants, in support of the numerical predictions and theimaged Q functions.To conclude, our experimental results are consistent with

the numerically calculated intracavity Wigner functionspresented in Fig. 4. While the different orientations of thesqueezed ellipse relative to the axes is a consequence of

(a)

(b)

FIG. 12. Numerical calculation of representative third-order (a)and fourth-order (b) cumulants of the filtered output field. Thedashed curves are for the monochromatic current pump (cubicand quartic correction), while the solid curves are for the Kerrquartic correction (bichromatic current pump or monochromaticflux pump). The third-order cumulant is exactly zero in thequartic correction case. The filter is a Gaussian filter with 4-MHzbandwidth. In all calculations, we consider γ ¼ Δ ¼ 0 andκ=2π ¼ 50 MHz.

(a) (b)

(c)

(e)

(d)

FIG. 13. Raw phase-space images of the JPA measurementchain output (including HEMT noise) as determined via homo-dyne measurements for (a) 5 dB and (c) 24 dB of phase-preserving gain. Each histogram is formed of 256 × 256 pointswhere the axes indicate the homodyne quadrature voltages VX,VP (�400-mV range), and the color scale indicates the relativecount intensity for each bin. The corresponding JPA output fieldQ functions inferred via the Lucy-Richardson deconvolution areshown in (b) and (d). The deconvolved distributions are plottedon the same measurement voltage range so that the larger areas ofthe distributions in (a) and (c) reflect the relative contributionsof HEMT noise to those measurements. The magnitude of thecorresponding output field cumulants scaled via the gain cali-bration described in Appendix D are shown in panel (e).


054030-15

different pump phases, the different orientations relative tothe ellipse of the S-like nonideality for the numericallycomputed intracavity field and the measured output fieldare explained by input-output theory.

VII. CONCLUSION

In summary, the Kerr-type nonlinear correction is alimiting factor for the measurement quantum efficiency ofJPAs and the squeezing level of their output field. Thiscorrection also leads to non-Gaussian signatures observedboth in the intracavity field Wigner function and the fourth-order cumulants of the output field. Our combined numeri-cal and experimental results allow us to explain a broadrange of experimental observations, such as the smallerthan expected quantum efficiency of the JPA [17,29,31,32],the saturation and decrease of squeezing at high gain inJPAs [14,15,28,29], and non-Gaussian signatures of theoutput field [15,28]. In particular, this work presents anexperimental characterization of the output field of a flux-driven JPA, as well as a direct experimental imaging of thenon-Gaussian distortions of the output field. In addition,we derive and compare the higher-order corrections to theJPA Hamiltonian for three different pumping schemes. Ourwork shows that in addition to a Kerr-type quartic correc-tion, cubic terms in the Hamiltonian are important in thecase of the monochromatic current pump. These additionalcorrections lead to larger deviations from the expected DPAbehavior such as lower attainable squeezing levels andlarger non-Gaussian signatures.In short, our results indicate three pathways to improving

the performance of the JPA as a squeezer and amplifier.First, in the case of a JPA operated with a monochromaticcurrent pump, moving instead to a bichromatic current-pumping scheme eliminates cubic corrections, leading to agreatly increased maximal squeezing level and reducednon-Gaussian signatures. Remarkably, this improvementcan be obtained for the same circuit and parameters.Otherwise, at the cost of a small added circuit complexity,but using only a single drive, flux pumping offers similaradvantages. Second, our results illustrate clearly the valueof designing JPAs with small Kerr nonlinearities. This canbe obtained by using SQUID arrays to dilute the non-linearity [4,9] or adding additional linear inductance [36].Finally, our work emphasizes the phase sensitivity of theJPA in the high-gain regime and the importance of fullycharacterizing the phase and frequency dependence of thegain matrix in order to operate at the optimal phases andfrequencies where the effects of nonidealities are minimal.To complement the numerical approach considered in this

paper, a better understanding of the higher-order correctionsdiscussed might be obtained by considering analyticalperturbation theory techniques [63]. Preliminary resultsare promising [64]. Similar analysis for a Josephson-junction-based traveling-wave parametric amplifier couldhelp the current experimental and theoretical effort [65–69].

Finally, while higher-order corrections hinder the perfor-mance of the JPA for squeezing and amplification, it canbecome a feature for other applications of the JPA such asrobust cat-state preparation and stabilization [70], which canbe used for quantum computation [71] and quantumannealing [72].

ACKNOWLEDGMENTS

We thank A. A. Clerk, N. Didier, A. Kamal, and M. H.Devoret for fruitful discussions. This work is supported bythe Army Research Office under Grant No. W911NF-14-1-0078, FRQ-NT, and Natural Sciences and EngineeringResearch Council of Canada (NSERC). A.W. E. acknowl-edges support from the U.S. Department of Defensethrough the NDSEG fellowship program. Computationsare made on the supercomputer Mammouth Parallele IIfrom Universite de Sherbrooke, managed by Calcul Quebecand Compute Canada. The operation of this supercomputeris funded by the Canada Foundation for Innovation,NanoQuebec, RMGA, and the Fonds de recherche duQuebec—Nature et technologies. This research is under-taken thanks in part to funding from the Canada FirstResearch Excellence Fund.

APPENDIX A: SOLUTION TO THE DPA’SINPUT-OUTPUT EQUATIONS

For this work to be self-contained, we present thestandard solution to the equations of motion of the DPAdiscussed in Sec. II [27,42]. Starting from Eq. (3), andintroducing the vectors

a¼�

a

a†

�; ain¼

aina†in

!; and bin¼

binb†in

!; ðA1Þ

the DPA can be described by the system of linear equations

_a ¼ Maþ ffiffiffiκ

pain þ

ffiffiffiγ

pbin; ðA2Þ

where the matrix M is

M ¼

264−�iΔþ κ

2

�−iλ

iλ��iΔ − κ

2

�375: ðA3Þ

This system of linear equations is solved by introducing theFourier-transformed operator

a½ω� ¼Z

∞

−∞dt eiωtaðtÞ: ðA4Þ

We note that a†½ω�, the Fourier transform of the creation

operator is related to the adjoint of a½ω� by a†½ω� ¼ a†½−ω�.Thus, the solution to the linearized equations of motion inFourier space is [27,42]


054030-16

a½ω� ¼ −ðMþ iω1Þ−1ð ffiffiffiκ

pain½ω� þ ffiffiffi

γp

bin½ω�Þ: ðA5Þ

Using the input-output boundary condition Eq. (4), thisresult leads to Eq. (5) of the main text.For completeness, we obtain from this expression the

general expressions for the second-order centered momentsof the intracavity field used in Sec. IV,

NDPA ¼ jλj22ðΔ2 þ κ2=4 − jλj2Þ ; ðA6Þ

MDPA ¼ −λðΔþ iκ=2Þ2ðΔ2 þ κ2=4 − jλj2Þ : ðA7Þ

Using the input-output boundary condition Eq. (4), theoutput field spectrums are [42]

Nout½ω� ¼jλj2κκ

ðΔ2 þ κ2=4 − ω2 − jλj2Þ2 þ κ2ω2; ðA8Þ

Mout½ω� ¼−iκλ½ðκ − iΔÞ2 þ ω2 þ jλj2�

ðΔ2 þ κ2=4 − ω2 − jλj2Þ2 þ κ2ω2; ðA9Þ

with Nout½ω� ¼Rdωha†out½ω�aout½ω�i=2π, and similarly,

Mout½ω� ¼Rdωhaout½ω�aout½ω�i=2π. These expressions

allow us to calculate analytically the DPA filtered outputfield spectrums presented in Sec. VI.

APPENDIX B: DETAILS OF THEDISPLACEMENT TRANSFORMATIONS

In this appendix, we detail the displacement transforma-tions used to derive the Hamiltonian Eq. (16) for themonochromatic current-pumped JPA and the HamiltonianEq. (22) for the bichromatic current-pumped JPA. Theunitary displacement transformation is [26]

DðβÞ ¼ expðβa† − β�aÞ; ðB1Þ

with D†ðβÞa DðβÞ ¼ aþ β, Dðβ1ÞDðβ2Þ ¼ Dðβ1 þ β2Þ,and β, βð1;2Þ scalar complex numbers.Applying this transformation on the driven Kerr

Hamiltonian

Hi ¼ Δa†aþ Λa†2a2 þ ϵðtÞa† þ ϵ�ðtÞa: ðB2Þ

With Δ ¼ ~ω0 − ωrot the detuning between the cavity andthe rotating-frame frequency, one obtains

H0i ¼ D†ðβÞH DðβÞ − i _DðβÞD†ðβÞ: ðB3Þ

Dropping constant terms and taking a → d to emphasizethe frame change, one obtains the Hamiltonian

H0i ¼ ðΔþ 4Λjβj2Þd†d

þ Λ½β2d†2 þ βd†2dþ H:c:� þ Λd†2d2; ðB4Þ

where the linear pump term is canceled by choosing thedisplacement parameter β such that

i _β ¼ ϵðtÞ þ�Δþ 2Λjβj2 − i

κ

2

�β: ðB5Þ

The additional term −iκβ=2 originates from applying thedisplacement transformation on the master equation (35)instead of only the Hamiltonian.In the single-current-pump case of Sec. III B, in a frame

rotating at the pump frequency (ωrot ¼ ωp) the pump issimply ϵðtÞ ¼ ϵ, and one obtains the results of the main textby taking α ¼ β.In the double-pump case of Sec. III C, in a frame rotating

at the average pump frequency Ω12 ¼ ðω1 þ ω2Þ=2, thedriving term is

ϵðtÞ ¼ ϵ1eþiΔ12t=2 þ ϵ2e−iΔ12t=2; ðB6Þ

with the detuning between the pumps Δ12 ¼ ω1 − ω2.Inserting Eq. (B6) in Eq. (B5) and taking the ansatzβ ¼ α1eþiΔ12t=2 þ α2e−iΔ12t=2 leads to the coupled nonlineardifferential equations

i _α1 ¼ ϵ1 þ�~ω0 − ω1 þ 2Λjα1j2 − i

κ

2

�α1

þ 2Λð2jα2j2 þ α1α�2e

−iΔ12tÞα1; ðB7Þ

i _α2 ¼ ϵ2 þ�~ω0 − ω2 þ 2Λjα2j2 − i

κ

2

�α2

þ 2Λð2jα1j2 þ α2α�1e

iΔ12tÞα2; ðB8Þ

and to the Hamiltonian of Eq. (22). For Δ12 ≫ 2jΛα1α2j,we can neglect the rotating terms under the RWA.Since the parametric resonance condition depends only

on the sum of the pump frequencies, and the amplitude ofthe classical field is bounded by the parametric threshold,one can choose Δ12 in order to enforce the validity of theRWA. Under this approximation, the equations are thesame as Eq. (18) in the monochromatic current-pump case,except that the equations are coupled through the frequencyshift induced by the cavity population at both pumpfrequencies.

APPENDIX C: EXPANSION OF THEFLUX-MODULATED JOSEPHSON ENERGY

In this appendix, we complement Sec. III D by givingthe analytical expressions for the Fourier coefficients ofthe Josephson energy of a flux modulated SQUID used inEq. (29). Using the Jacobi-Anger formula [73]


054030-17

expðix cos θÞ ¼ J0ðxÞ þ 2X∞n¼0

inJnðxÞ cos nθ; ðC1Þ

where JnðxÞ is the nth Bessel function of the first kind, oneobtains the Fourier coefficients

Eð0ÞJ ¼ EJJ0ðδfÞ cosF; ðC2Þ

Eð2n−1ÞJ ¼ 2EJð−1ÞnJ2n−1ðδfÞ sinF; ðC3Þ

Eð2nÞJ ¼ 2EJð−1ÞnJ2nðδfÞ cosF; ðC4Þ

with n ∈ f1; 2; 3…g.In the case of a small amplitude flux pump (δf ≪ 1), the

Bessel functions can be expanded. The leading term of eachcoefficient is such that

EðnÞJ ∝

1

n!

�δf2

�n: ðC5Þ

More explicitly, the first three coefficients of the Fourierexpansion are, to leading order in δf,

Eð0ÞJ ≈ EJ cosF; ðC6Þ

Eð1ÞJ ≈ −EJδf sinF; ðC7Þ

Eð2ÞJ ≈ −

EJδf2 cosF4

; ðC8Þ

in agreement with the expressions of Refs. [35,49].

APPENDIX D: EXPERIMENTAL SETUPAND METHODS

This appendix presents details of the experimental setupand techniques used to obtain the experimental data for aflux-driven JPA presented in Figs. 11 and 13.

1. Experimental setup

We focus our characterization on an aluminum, lumped-element JPA. This design is of particular interest, as it hasbeen widely adopted for superconducting qubit readout[8,13,17]. The device consists of a capacitance (3.2 pF)shunted by a SQUID [LJðΦ ¼ 0Þ ¼ 45 pH]. From simu-lations, we estimate the geometric inductance of our designis 35 pH leading to a participation ratio p ¼ 0.8. Fromthese values, we estimate a Kerr nonlinearity Λ=2π ¼−1.55 MHz and from gain-bandwidth measurements acavity damping rate κ=2π ¼ 130 MHz. This empiricalvalue of κ is lower than expected from the lumped-elementcapacitance; we attribute this difference to spatial imped-ance variations in the JPA environment [10]. The SQUIDis flux pumped to provide up to 25 dB of gain using an

on-chip bias line [5], and the device is mounted with a 180°hybrid launch to reject common mode noise. The amplifieris shielded by an aluminum box and mounted at the baseplate of a dilution refrigerator (approximately 20 mK). Thesignal from the JPA is further amplified with a commercialHEMT amplifier at 4 K and subsequent room-temperatureamplifiers before being down-converted and digitized in ahomodyne measurement.The homodyne measurement samples the values of the

conjugate quadrature components X and P described bythe complex quadrature operator S ¼ X þ iP [59]. For eachmeasurement, we average 256 consecutive voltage samplesacquired at 1 GS=s, effectively filtering S so as to analyzeonly highly squeezed spectral components near the cavityfrequency. The flux pump at frequency 2ω is generatedusing a frequency doubler and the same microwave sourcethat produces the local oscillator tone for the homodynesetup, allowing for good phase stability over the course ofextended measurements. In addition, the JPA is hooked upto a switch on the base stage of the dilution refrigerator. Theother port of the switch connects to a qubit dispersivelycoupled to a tunable resonator [74]. Through measurementsof the qubit Stark shift, this setup enables a precisecalibration of the power gain between the JPA and theanalog-to-digital converter (ADC) [69], a necessary inputfor our analysis.

2. Single-path reconstruction method

To characterize the properties of the output field, weutilize the single-path reconstruction method developedby Eichler et al. [58]. We construct two histograms of S,corresponding to the distribution with the JPA pump on,D½ρ�ðSÞ, and to the distribution with the JPA pump off,D½j0ih0j�ðSÞ. We interleave the two measurements to mitigatethe effects of experimental drift. We analyze the momentsof the two histograms hðS†ÞnSmiρ and hðS†ÞnSmij0ih0j toinfer the normally ordered moments of the output fieldhða†Þnami through the expressions

hðS†ÞnSmiρ ¼ GðnþmÞ=2c

Xn;mi;j¼0

�mj

��ni

�hða†Þiaji

× hhn−iðh†Þm−ji ðD1Þ

and

hðS†ÞnSmij0ih0j ¼ GðnþmÞ=2c hhnðh†Þmi: ðD2Þ

Here, Gc is the gain of the measurement chain betweenthe JPA output and the ADC in the photon-number basis[61,75], and h is an effective noise mode dominated bythe added noise of the HEMTamplifier. Experimentally, thepower gain of the measurement chain is measured to bebetween 100.1 and 100.5 dB. These bounds on the power


054030-18

gain are reflected in the systematic uncertainty in thesqueezing levels presented in Fig. 11. Note thatEq. (D2) is a limiting case of Eq. (D1) when the JPApump is off and a is in the vacuum state. We iterativelysolve these equations to compute the moments of the JPAoutput field and also quantify the cumulants of the outputfield denoted by ⟪ða†Þnam⟫. For Gaussian states, suchas ideal squeezed states, the cumulants are zero fornþm > 2. At each gain setting, we histogram 108 noisemeasurements, which provides sufficient resolution to inferthe field moments up to fourth order.

3. Imaging the output field using deconvolutions

To directly image the distortions of the output field dueto nonidealities, we implement a complementary analysistechnique based on applying a series of deconvolutions toD½ρ�ðSÞ and D½j0ih0j�ðSÞ. This method relies on the fact thatthese discrete distributions can be approximated as aconvolution of quasiprobability distributions for the JPAoutput field and an effective noise mode of the measure-ment chain [76]:

D½ρ�ðSÞ ≈ 1

GcðQJPA⋆PHEMTÞ

�αffiffiffiffiffiffiGc

p�

ðD3Þ

and

D½j0ih0j�ðSÞ ≈ 1

GcðQvacuum⋆PHEMTÞ

�αffiffiffiffiffiffiGc

p�: ðD4Þ

Here, QJPA, Qvacuum, and PHEMT are Husimi Q andGlauber-Sudarshan P representations that describe theJPA output field, the vacuum state, and the noise mode[77]. Given that the Q function can be obtained from theWigner function via a Gaussian smoothing filter, we expectthe distortions imaged by this deconvolution technique tobe characteristic of both phase-space representations.To reconstruct the output field, we first deconvolve

D½j0ih0j�ðSÞ with Qvacuum to obtain ð1=GcÞPHEMT and thendeconvolve D½ρ�ðSÞ with ð1=GcÞPHEMT to obtain QJPA.Both deconvolutions are performed in MATLAB using theLucy-Richardson method [78]. For the measurements inFig. 13, the power gain of the measurement chain isincreased by 5 dB relative to the bounds quoted inAppendix D 2. We empirically find this larger gain leadsto improved resolution of the low-density tails of the outputfield Q functions. In all cases, inferences of the output fieldcumulants obtained via the moment-based reconstructionagree with inferences of the cumulants calculated from theQ functions.

APPENDIX E: NUMERICAL CALCULATIONOF FILTERED MOMENTS

In Sec. VI, we calculate moments of the filtered outputfield. Here, we give the details of the numerical technique

used to obtain the results presented there. As an illustrationof the technique, we consider the third-order moment

hD†D2i ¼ZZZ

∞

−∞dt1dt2dt3fð−t1Þfð−t2Þ

× fð−t3ÞMt1;t2;t3 ; ðE1Þ

with the three-time correlation function

Mt1;t2;t3 ¼ ha†outðt1Þaoutðt2Þaoutðt3Þi¼ κ3=2ha†ðt1Þaðt2Þaðt3Þi; ðE2Þ

where the last equality is valid for a vacuum inputfield [79].In order to use the quantum regression formula, we

separate the integral in a sum over all possible timeorderings

hD†D2i ¼ZZ

∞

0

dt1dt2Ft1;t2 ½Mt1þt2;t1;0

þMt1;t1þt2;0 þM0;t1þt2;t1 �; ðE3Þ

where using the invariance under time translation for asteady state and assuming a purely real filter, we define

Ft1;t2 ¼Z

∞

−∞dt3fð−t1 − t2 − t3Þfð−t2 − t3Þfð−t3Þ: ðE4Þ

Finally, we integrate over each correlation function thatwe calculate using the general formula of the quantumregression result [27]. For the three-time correlation func-tion, we obtain

Mτ;t1;0 ¼ Trfa†outVτ;t1 ½aoutVt1;0ðaoutρssÞ�g;Mt1;τ;0 ¼ TrfaoutVτ;t1 ½Vt1;0ðaoutρssÞa†out�g;M0;τ;t1 ¼ TrfaoutVτ;t1 ½aoutVt1;0ðρssa†outÞ�g; ðE5Þ

with Vðt1; 0Þ the evolution superoperator from t ¼ 0 tot ¼ t1, and τ ¼ t1 þ t2. Numerically, the evolution isperformed by integrating the master equation using theRunge-Kutta solver of the GSL numerical library [80].

APPENDIX F: SQUEEZING LEVEL OF THEFILTERED DPA OUTPUT FIELD

In addition to the numerical approach of Appendix E,the results of Figs. 10 and 11 for the DPA can be obtainedmore simply by integrating the analytical expressions ofEqs. (A8) and (A9) using

hD†Di ¼Z

∞

−∞dωjf½ω�j2Nout½ω� ¼ Nf; ðF1Þ


054030-19

hD2i ¼Z

∞

−∞dωjf½ω�j2Mout½ω� ¼ Mf; ðF2Þ

with f½ω� the Fourier transform of a real-time-domain filterfunction fðtÞ. In particular, in the case of Fig. 10 whereκ ¼ κ and Δ ¼ 0, the minimal variance of the filteredoutput field is

hΔX2mini ¼

1

2

Zdωjf½ω�j2

�1 −

2κjλjðκ=2þ jλjÞ2 þ ω2

�:

ðF3ÞAs we mention in the main text, the integrand is minimal(maximal squeezing) at the center frequency ω ¼ 0 andincreases (reduced squeezing) away from this frequency.Thus, as we discuss in Sec. VI B, the minimal variance ofthe filtered field will always be equal to or larger than thevariance at the center frequency. This limits the squeezinglevel of the JPA filtered output field, even without anynonidealities.To further illustrate this effect, we consider the case

of a narrow-band filter centered at frequency ω0 such thatjf½ω�j2 ≈ δðω − ω0Þ. In that case, the squeezing level is

Sfðω0Þ ¼ 1þ 2κjλjðκ=2 − jλjÞ2 þ ω2

0

; ðF4Þ

which diverges only for ω0 ¼ 0 and otherwise reaches afinite maximal value of 1þ κ2=ω2

0 even at the parametricthreshold. Neglecting the constant background, the halfwidth at half maximum of Sfðω0Þ is κ=2 − jλj, whichdecreases with increasing gain reaching zero at the para-metric threshold. This relation is a different illustration ofthe gain-bandwidth trade-off in cavity-based parametricamplifiers [60].

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